sysdeps/libm-ieee754/e_j0.c

   1 /* @(#)e_j0.c 5.1 93/09/24 */
   2 /*
   3  * ====================================================
   4  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   5  *
   6  * Developed at SunPro, a Sun Microsystems, Inc. business.
   7  * Permission to use, copy, modify, and distribute this
   8  * software is freely granted, provided that this notice
   9  * is preserved.
  10  * ====================================================
  11  */
  12 /* Modified by Naohiko Shimizu/Tokai University, Japan 1997/08/26,
  13    for performance improvement on pipelined processors.
  14 */
  15
  16 #if defined(LIBM_SCCS) && !defined(lint)
  17 static char rcsid[] = "$NetBSD: e_j0.c,v 1.8 1995/05/10 20:45:23 jtc Exp $";
  18 #endif
  19
  20 /* __ieee754_j0(x), __ieee754_y0(x)
  21  * Bessel function of the first and second kinds of order zero.
  22  * Method -- j0(x):
  23  *      1. For tiny x, we use j0(x) = 1 - x^2/4 + x^4/64 - ...
  24  *      2. Reduce x to |x| since j0(x)=j0(-x),  and
  25  *         for x in (0,2)
  26  *              j0(x) = 1-z/4+ z^2*R0/S0,  where z = x*x;
  27  *         (precision:  |j0-1+z/4-z^2R0/S0 |<2**-63.67 )
  28  *         for x in (2,inf)
  29  *              j0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)-q0(x)*sin(x0))
  30  *         where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
  31  *         as follow:
  32  *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
  33  *                      = 1/sqrt(2) * (cos(x) + sin(x))
  34  *              sin(x0) = sin(x)cos(pi/4)-cos(x)sin(pi/4)
  35  *                      = 1/sqrt(2) * (sin(x) - cos(x))
  36  *         (To avoid cancellation, use
  37  *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
  38  *          to compute the worse one.)
  39  *
  40  *      3 Special cases
  41  *              j0(nan)= nan
  42  *              j0(0) = 1
  43  *              j0(inf) = 0
  44  *
  45  * Method -- y0(x):
  46  *      1. For x<2.
  47  *         Since
  48  *              y0(x) = 2/pi*(j0(x)*(ln(x/2)+Euler) + x^2/4 - ...)
  49  *         therefore y0(x)-2/pi*j0(x)*ln(x) is an even function.
  50  *         We use the following function to approximate y0,
  51  *              y0(x) = U(z)/V(z) + (2/pi)*(j0(x)*ln(x)), z= x^2
  52  *         where
  53  *              U(z) = u00 + u01*z + ... + u06*z^6
  54  *              V(z) = 1  + v01*z + ... + v04*z^4
  55  *         with absolute approximation error bounded by 2**-72.
  56  *         Note: For tiny x, U/V = u0 and j0(x)~1, hence
  57  *              y0(tiny) = u0 + (2/pi)*ln(tiny), (choose tiny<2**-27)
  58  *      2. For x>=2.
  59  *              y0(x) = sqrt(2/(pi*x))*(p0(x)*cos(x0)+q0(x)*sin(x0))
  60  *         where x0 = x-pi/4. It is better to compute sin(x0),cos(x0)
  61  *         by the method mentioned above.
  62  *      3. Special cases: y0(0)=-inf, y0(x<0)=NaN, y0(inf)=0.
  63  */
  64
  65 #include "math.h"
  66 #include "math_private.h"
  67
  68 #ifdef __STDC__
  69 static double pzero(double), qzero(double);
  70 #else
  71 static double pzero(), qzero();
  72 #endif
  73
  74 #ifdef __STDC__
  75 static const double
  76 #else
  77 static double
  78 #endif
  79 huge    = 1e300,
  80 one     = 1.0,
  81 invsqrtpi=  5.64189583547756279280e-01, /* 0x3FE20DD7, 0x50429B6D */
  82 tpi      =  6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
  83                 /* R0/S0 on [0, 2.00] */
  84 R[]  =  {0.0, 0.0, 1.56249999999999947958e-02, /* 0x3F8FFFFF, 0xFFFFFFFD */
  85  -1.89979294238854721751e-04, /* 0xBF28E6A5, 0xB61AC6E9 */
  86   1.82954049532700665670e-06, /* 0x3EBEB1D1, 0x0C503919 */
  87  -4.61832688532103189199e-09}, /* 0xBE33D5E7, 0x73D63FCE */
  88 S[]  =  {0.0, 1.56191029464890010492e-02, /* 0x3F8FFCE8, 0x82C8C2A4 */
  89   1.16926784663337450260e-04, /* 0x3F1EA6D2, 0xDD57DBF4 */
  90   5.13546550207318111446e-07, /* 0x3EA13B54, 0xCE84D5A9 */
  91   1.16614003333790000205e-09}; /* 0x3E1408BC, 0xF4745D8F */
  92
  93 #ifdef __STDC__
  94 static const double zero = 0.0;
  95 #else
  96 static double zero = 0.0;
  97 #endif
  98
  99 #ifdef __STDC__
 100         double __ieee754_j0(double x)
 101 #else
 102         double __ieee754_j0(x)
 103         double x;
 104 #endif
 105 {
 106         double z, s,c,ss,cc,r,u,v,r1,r2,s1,s2,z2,z4;
 107         int32_t hx,ix;
 108
 109         GET_HIGH_WORD(hx,x);
 110         ix = hx&0x7fffffff;
 111         if(ix>=0x7ff00000) return one/(x*x);
 112         x = fabs(x);
 113         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
 114                 s = __sin(x);
 115                 c = __cos(x);
 116                 ss = s-c;
 117                 cc = s+c;
 118                 if(ix<0x7fe00000) {  /* make sure x+x not overflow */
 119                     z = -__cos(x+x);
 120                     if ((s*c)<zero) cc = z/ss;
 121                     else            ss = z/cc;
 122                 }
 123         /*
 124          * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
 125          * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
 126          */
 127                 if(ix>0x48000000) z = (invsqrtpi*cc)/__sqrt(x);
 128                 else {
 129                     u = pzero(x); v = qzero(x);
 130                     z = invsqrtpi*(u*cc-v*ss)/__sqrt(x);
 131                 }
 132                 return z;
 133         }
 134         if(ix<0x3f200000) {     /* |x| < 2**-13 */
 135             if(huge+x>one) {    /* raise inexact if x != 0 */
 136                 if(ix<0x3e400000) return one;   /* |x|<2**-27 */
 137                 else          return one - 0.25*x*x;
 138             }
 139         }
 140         z = x*x;
 141 #ifdef DO_NOT_USE_THIS
 142         r =  z*(R02+z*(R03+z*(R04+z*R05)));
 143         s =  one+z*(S01+z*(S02+z*(S03+z*S04)));
 144 #else
 145         r1 = z*R[2]; z2=z*z;
 146         r2 = R[3]+z*R[4]; z4=z2*z2;
 147         r  = r1 + z2*r2 + z4*R[5];
 148         s1 = one+z*S[1];
 149         s2 = S[2]+z*S[3];
 150         s = s1 + z2*s2 + z4*S[4];
 151 #endif
 152         if(ix < 0x3FF00000) {   /* |x| < 1.00 */
 153             return one + z*(-0.25+(r/s));
 154         } else {
 155             u = 0.5*x;
 156             return((one+u)*(one-u)+z*(r/s));
 157         }
 158 }
 159
 160 #ifdef __STDC__
 161 static const double
 162 #else
 163 static double
 164 #endif
 165 U[]  = {-7.38042951086872317523e-02, /* 0xBFB2E4D6, 0x99CBD01F */
 166   1.76666452509181115538e-01, /* 0x3FC69D01, 0x9DE9E3FC */
 167  -1.38185671945596898896e-02, /* 0xBF8C4CE8, 0xB16CFA97 */
 168   3.47453432093683650238e-04, /* 0x3F36C54D, 0x20B29B6B */
 169  -3.81407053724364161125e-06, /* 0xBECFFEA7, 0x73D25CAD */
 170   1.95590137035022920206e-08, /* 0x3E550057, 0x3B4EABD4 */
 171  -3.98205194132103398453e-11}, /* 0xBDC5E43D, 0x693FB3C8 */
 172 V[]  =  {1.27304834834123699328e-02, /* 0x3F8A1270, 0x91C9C71A */
 173   7.60068627350353253702e-05, /* 0x3F13ECBB, 0xF578C6C1 */
 174   2.59150851840457805467e-07, /* 0x3E91642D, 0x7FF202FD */
 175   4.41110311332675467403e-10}; /* 0x3DFE5018, 0x3BD6D9EF */
 176
 177 #ifdef __STDC__
 178         double __ieee754_y0(double x)
 179 #else
 180         double __ieee754_y0(x)
 181         double x;
 182 #endif
 183 {
 184         double z, s,c,ss,cc,u,v,z2,z4,z6,u1,u2,u3,v1,v2;
 185         int32_t hx,ix,lx;
 186
 187         EXTRACT_WORDS(hx,lx,x);
 188         ix = 0x7fffffff&hx;
 189     /* Y0(NaN) is NaN, y0(-inf) is Nan, y0(inf) is 0  */
 190         if(ix>=0x7ff00000) return  one/(x+x*x);
 191         if((ix|lx)==0) return -one/zero;
 192         if(hx<0) return zero/zero;
 193         if(ix >= 0x40000000) {  /* |x| >= 2.0 */
 194         /* y0(x) = sqrt(2/(pi*x))*(p0(x)*sin(x0)+q0(x)*cos(x0))
 195          * where x0 = x-pi/4
 196          *      Better formula:
 197          *              cos(x0) = cos(x)cos(pi/4)+sin(x)sin(pi/4)
 198          *                      =  1/sqrt(2) * (sin(x) + cos(x))
 199          *              sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
 200          *                      =  1/sqrt(2) * (sin(x) - cos(x))
 201          * To avoid cancellation, use
 202          *              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
 203          * to compute the worse one.
 204          */
 205                 s = __sin(x);
 206                 c = __cos(x);
 207                 ss = s-c;
 208                 cc = s+c;
 209         /*
 210          * j0(x) = 1/sqrt(pi) * (P(0,x)*cc - Q(0,x)*ss) / sqrt(x)
 211          * y0(x) = 1/sqrt(pi) * (P(0,x)*ss + Q(0,x)*cc) / sqrt(x)
 212          */
 213                 if(ix<0x7fe00000) {  /* make sure x+x not overflow */
 214                     z = -__cos(x+x);
 215                     if ((s*c)<zero) cc = z/ss;
 216                     else            ss = z/cc;
 217                 }
 218                 if(ix>0x48000000) z = (invsqrtpi*ss)/__sqrt(x);
 219                 else {
 220                     u = pzero(x); v = qzero(x);
 221                     z = invsqrtpi*(u*ss+v*cc)/__sqrt(x);
 222                 }
 223                 return z;
 224         }
 225         if(ix<=0x3e400000) {    /* x < 2**-27 */
 226             return(U[0] + tpi*__ieee754_log(x));
 227         }
 228         z = x*x;
 229 #ifdef DO_NOT_USE_THIS
 230         u = u00+z*(u01+z*(u02+z*(u03+z*(u04+z*(u05+z*u06)))));
 231         v = one+z*(v01+z*(v02+z*(v03+z*v04)));
 232 #else
 233         u1 = U[0]+z*U[1]; z2=z*z;
 234         u2 = U[2]+z*U[3]; z4=z2*z2;
 235         u3 = U[4]+z*U[5]; z6=z4*z2;
 236         u = u1 + z2*u2 + z4*u3 + z6*U[6];
 237         v1 = one+z*V[1];
 238         v2 = V[2]+z*V[3];
 239         v = v1 + z2*v2 + z4*V[4];
 240 #endif
 241         return(u/v + tpi*(__ieee754_j0(x)*__ieee754_log(x)));
 242 }
 243
 244 /* The asymptotic expansions of pzero is
 245  *      1 - 9/128 s^2 + 11025/98304 s^4 - ...,  where s = 1/x.
 246  * For x >= 2, We approximate pzero by
 247  *      pzero(x) = 1 + (R/S)
 248  * where  R = pR0 + pR1*s^2 + pR2*s^4 + ... + pR5*s^10
 249  *        S = 1 + pS0*s^2 + ... + pS4*s^10
 250  * and
 251  *      | pzero(x)-1-R/S | <= 2  ** ( -60.26)
 252  */
 253 #ifdef __STDC__
 254 static const double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 255 #else
 256 static double pR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 257 #endif
 258   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
 259  -7.03124999999900357484e-02, /* 0xBFB1FFFF, 0xFFFFFD32 */
 260  -8.08167041275349795626e+00, /* 0xC02029D0, 0xB44FA779 */
 261  -2.57063105679704847262e+02, /* 0xC0701102, 0x7B19E863 */
 262  -2.48521641009428822144e+03, /* 0xC0A36A6E, 0xCD4DCAFC */
 263  -5.25304380490729545272e+03, /* 0xC0B4850B, 0x36CC643D */
 264 };
 265 #ifdef __STDC__
 266 static const double pS8[5] = {
 267 #else
 268 static double pS8[5] = {
 269 #endif
 270   1.16534364619668181717e+02, /* 0x405D2233, 0x07A96751 */
 271   3.83374475364121826715e+03, /* 0x40ADF37D, 0x50596938 */
 272   4.05978572648472545552e+04, /* 0x40E3D2BB, 0x6EB6B05F */
 273   1.16752972564375915681e+05, /* 0x40FC810F, 0x8F9FA9BD */
 274   4.76277284146730962675e+04, /* 0x40E74177, 0x4F2C49DC */
 275 };
 276
 277 #ifdef __STDC__
 278 static const double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 279 #else
 280 static double pR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 281 #endif
 282  -1.14125464691894502584e-11, /* 0xBDA918B1, 0x47E495CC */
 283  -7.03124940873599280078e-02, /* 0xBFB1FFFF, 0xE69AFBC6 */
 284  -4.15961064470587782438e+00, /* 0xC010A370, 0xF90C6BBF */
 285  -6.76747652265167261021e+01, /* 0xC050EB2F, 0x5A7D1783 */
 286  -3.31231299649172967747e+02, /* 0xC074B3B3, 0x6742CC63 */
 287  -3.46433388365604912451e+02, /* 0xC075A6EF, 0x28A38BD7 */
 288 };
 289 #ifdef __STDC__
 290 static const double pS5[5] = {
 291 #else
 292 static double pS5[5] = {
 293 #endif
 294   6.07539382692300335975e+01, /* 0x404E6081, 0x0C98C5DE */
 295   1.05125230595704579173e+03, /* 0x40906D02, 0x5C7E2864 */
 296   5.97897094333855784498e+03, /* 0x40B75AF8, 0x8FBE1D60 */
 297   9.62544514357774460223e+03, /* 0x40C2CCB8, 0xFA76FA38 */
 298   2.40605815922939109441e+03, /* 0x40A2CC1D, 0xC70BE864 */
 299 };
 300
 301 #ifdef __STDC__
 302 static const double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 303 #else
 304 static double pR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 305 #endif
 306  -2.54704601771951915620e-09, /* 0xBE25E103, 0x6FE1AA86 */
 307  -7.03119616381481654654e-02, /* 0xBFB1FFF6, 0xF7C0E24B */
 308  -2.40903221549529611423e+00, /* 0xC00345B2, 0xAEA48074 */
 309  -2.19659774734883086467e+01, /* 0xC035F74A, 0x4CB94E14 */
 310  -5.80791704701737572236e+01, /* 0xC04D0A22, 0x420A1A45 */
 311  -3.14479470594888503854e+01, /* 0xC03F72AC, 0xA892D80F */
 312 };
 313 #ifdef __STDC__
 314 static const double pS3[5] = {
 315 #else
 316 static double pS3[5] = {
 317 #endif
 318   3.58560338055209726349e+01, /* 0x4041ED92, 0x84077DD3 */
 319   3.61513983050303863820e+02, /* 0x40769839, 0x464A7C0E */
 320   1.19360783792111533330e+03, /* 0x4092A66E, 0x6D1061D6 */
 321   1.12799679856907414432e+03, /* 0x40919FFC, 0xB8C39B7E */
 322   1.73580930813335754692e+02, /* 0x4065B296, 0xFC379081 */
 323 };
 324
 325 #ifdef __STDC__
 326 static const double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 327 #else
 328 static double pR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 329 #endif
 330  -8.87534333032526411254e-08, /* 0xBE77D316, 0xE927026D */
 331  -7.03030995483624743247e-02, /* 0xBFB1FF62, 0x495E1E42 */
 332  -1.45073846780952986357e+00, /* 0xBFF73639, 0x8A24A843 */
 333  -7.63569613823527770791e+00, /* 0xC01E8AF3, 0xEDAFA7F3 */
 334  -1.11931668860356747786e+01, /* 0xC02662E6, 0xC5246303 */
 335  -3.23364579351335335033e+00, /* 0xC009DE81, 0xAF8FE70F */
 336 };
 337 #ifdef __STDC__
 338 static const double pS2[5] = {
 339 #else
 340 static double pS2[5] = {
 341 #endif
 342   2.22202997532088808441e+01, /* 0x40363865, 0x908B5959 */
 343   1.36206794218215208048e+02, /* 0x4061069E, 0x0EE8878F */
 344   2.70470278658083486789e+02, /* 0x4070E786, 0x42EA079B */
 345   1.53875394208320329881e+02, /* 0x40633C03, 0x3AB6FAFF */
 346   1.46576176948256193810e+01, /* 0x402D50B3, 0x44391809 */
 347 };
 348
 349 #ifdef __STDC__
 350         static double pzero(double x)
 351 #else
 352         static double pzero(x)
 353         double x;
 354 #endif
 355 {
 356 #ifdef __STDC__
 357         const double *p,*q;
 358 #else
 359         double *p,*q;
 360 #endif
 361         double z,r,s,z2,z4,r1,r2,r3,s1,s2,s3;
 362         int32_t ix;
 363         GET_HIGH_WORD(ix,x);
 364         ix &= 0x7fffffff;
 365         if(ix>=0x40200000)     {p = pR8; q= pS8;}
 366         else if(ix>=0x40122E8B){p = pR5; q= pS5;}
 367         else if(ix>=0x4006DB6D){p = pR3; q= pS3;}
 368         else if(ix>=0x40000000){p = pR2; q= pS2;}
 369         z = one/(x*x);
 370 #ifdef DO_NOT_USE_THIS
 371         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 372         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))));
 373 #else
 374         r1 = p[0]+z*p[1]; z2=z*z;
 375         r2 = p[2]+z*p[3]; z4=z2*z2;
 376         r3 = p[4]+z*p[5];
 377         r = r1 + z2*r2 + z4*r3;
 378         s1 = one+z*q[0];
 379         s2 = q[1]+z*q[2];
 380         s3 = q[3]+z*q[4];
 381         s = s1 + z2*s2 + z4*s3;
 382 #endif
 383         return one+ r/s;
 384 }
 385
 386
 387 /* For x >= 8, the asymptotic expansions of qzero is
 388  *      -1/8 s + 75/1024 s^3 - ..., where s = 1/x.
 389  * We approximate pzero by
 390  *      qzero(x) = s*(-1.25 + (R/S))
 391  * where  R = qR0 + qR1*s^2 + qR2*s^4 + ... + qR5*s^10
 392  *        S = 1 + qS0*s^2 + ... + qS5*s^12
 393  * and
 394  *      | qzero(x)/s +1.25-R/S | <= 2  ** ( -61.22)
 395  */
 396 #ifdef __STDC__
 397 static const double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 398 #else
 399 static double qR8[6] = { /* for x in [inf, 8]=1/[0,0.125] */
 400 #endif
 401   0.00000000000000000000e+00, /* 0x00000000, 0x00000000 */
 402   7.32421874999935051953e-02, /* 0x3FB2BFFF, 0xFFFFFE2C */
 403   1.17682064682252693899e+01, /* 0x40278952, 0x5BB334D6 */
 404   5.57673380256401856059e+02, /* 0x40816D63, 0x15301825 */
 405   8.85919720756468632317e+03, /* 0x40C14D99, 0x3E18F46D */
 406   3.70146267776887834771e+04, /* 0x40E212D4, 0x0E901566 */
 407 };
 408 #ifdef __STDC__
 409 static const double qS8[6] = {
 410 #else
 411 static double qS8[6] = {
 412 #endif
 413   1.63776026895689824414e+02, /* 0x406478D5, 0x365B39BC */
 414   8.09834494656449805916e+03, /* 0x40BFA258, 0x4E6B0563 */
 415   1.42538291419120476348e+05, /* 0x41016652, 0x54D38C3F */
 416   8.03309257119514397345e+05, /* 0x412883DA, 0x83A52B43 */
 417   8.40501579819060512818e+05, /* 0x4129A66B, 0x28DE0B3D */
 418  -3.43899293537866615225e+05, /* 0xC114FD6D, 0x2C9530C5 */
 419 };
 420
 421 #ifdef __STDC__
 422 static const double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 423 #else
 424 static double qR5[6] = { /* for x in [8,4.5454]=1/[0.125,0.22001] */
 425 #endif
 426   1.84085963594515531381e-11, /* 0x3DB43D8F, 0x29CC8CD9 */
 427   7.32421766612684765896e-02, /* 0x3FB2BFFF, 0xD172B04C */
 428   5.83563508962056953777e+00, /* 0x401757B0, 0xB9953DD3 */
 429   1.35111577286449829671e+02, /* 0x4060E392, 0x0A8788E9 */
 430   1.02724376596164097464e+03, /* 0x40900CF9, 0x9DC8C481 */
 431   1.98997785864605384631e+03, /* 0x409F17E9, 0x53C6E3A6 */
 432 };
 433 #ifdef __STDC__
 434 static const double qS5[6] = {
 435 #else
 436 static double qS5[6] = {
 437 #endif
 438   8.27766102236537761883e+01, /* 0x4054B1B3, 0xFB5E1543 */
 439   2.07781416421392987104e+03, /* 0x40A03BA0, 0xDA21C0CE */
 440   1.88472887785718085070e+04, /* 0x40D267D2, 0x7B591E6D */
 441   5.67511122894947329769e+04, /* 0x40EBB5E3, 0x97E02372 */
 442   3.59767538425114471465e+04, /* 0x40E19118, 0x1F7A54A0 */
 443  -5.35434275601944773371e+03, /* 0xC0B4EA57, 0xBEDBC609 */
 444 };
 445
 446 #ifdef __STDC__
 447 static const double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 448 #else
 449 static double qR3[6] = {/* for x in [4.547,2.8571]=1/[0.2199,0.35001] */
 450 #endif
 451   4.37741014089738620906e-09, /* 0x3E32CD03, 0x6ADECB82 */
 452   7.32411180042911447163e-02, /* 0x3FB2BFEE, 0x0E8D0842 */
 453   3.34423137516170720929e+00, /* 0x400AC0FC, 0x61149CF5 */
 454   4.26218440745412650017e+01, /* 0x40454F98, 0x962DAEDD */
 455   1.70808091340565596283e+02, /* 0x406559DB, 0xE25EFD1F */
 456   1.66733948696651168575e+02, /* 0x4064D77C, 0x81FA21E0 */
 457 };
 458 #ifdef __STDC__
 459 static const double qS3[6] = {
 460 #else
 461 static double qS3[6] = {
 462 #endif
 463   4.87588729724587182091e+01, /* 0x40486122, 0xBFE343A6 */
 464   7.09689221056606015736e+02, /* 0x40862D83, 0x86544EB3 */
 465   3.70414822620111362994e+03, /* 0x40ACF04B, 0xE44DFC63 */
 466   6.46042516752568917582e+03, /* 0x40B93C6C, 0xD7C76A28 */
 467   2.51633368920368957333e+03, /* 0x40A3A8AA, 0xD94FB1C0 */
 468  -1.49247451836156386662e+02, /* 0xC062A7EB, 0x201CF40F */
 469 };
 470
 471 #ifdef __STDC__
 472 static const double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 473 #else
 474 static double qR2[6] = {/* for x in [2.8570,2]=1/[0.3499,0.5] */
 475 #endif
 476   1.50444444886983272379e-07, /* 0x3E84313B, 0x54F76BDB */
 477   7.32234265963079278272e-02, /* 0x3FB2BEC5, 0x3E883E34 */
 478   1.99819174093815998816e+00, /* 0x3FFFF897, 0xE727779C */
 479   1.44956029347885735348e+01, /* 0x402CFDBF, 0xAAF96FE5 */
 480   3.16662317504781540833e+01, /* 0x403FAA8E, 0x29FBDC4A */
 481   1.62527075710929267416e+01, /* 0x403040B1, 0x71814BB4 */
 482 };
 483 #ifdef __STDC__
 484 static const double qS2[6] = {
 485 #else
 486 static double qS2[6] = {
 487 #endif
 488   3.03655848355219184498e+01, /* 0x403E5D96, 0xF7C07AED */
 489   2.69348118608049844624e+02, /* 0x4070D591, 0xE4D14B40 */
 490   8.44783757595320139444e+02, /* 0x408A6645, 0x22B3BF22 */
 491   8.82935845112488550512e+02, /* 0x408B977C, 0x9C5CC214 */
 492   2.12666388511798828631e+02, /* 0x406A9553, 0x0E001365 */
 493  -5.31095493882666946917e+00, /* 0xC0153E6A, 0xF8B32931 */
 494 };
 495
 496 #ifdef __STDC__
 497         static double qzero(double x)
 498 #else
 499         static double qzero(x)
 500         double x;
 501 #endif
 502 {
 503 #ifdef __STDC__
 504         const double *p,*q;
 505 #else
 506         double *p,*q;
 507 #endif
 508         double s,r,z,z2,z4,z6,r1,r2,r3,s1,s2,s3;
 509         int32_t ix;
 510         GET_HIGH_WORD(ix,x);
 511         ix &= 0x7fffffff;
 512         if(ix>=0x40200000)     {p = qR8; q= qS8;}
 513         else if(ix>=0x40122E8B){p = qR5; q= qS5;}
 514         else if(ix>=0x4006DB6D){p = qR3; q= qS3;}
 515         else if(ix>=0x40000000){p = qR2; q= qS2;}
 516         z = one/(x*x);
 517 #ifdef DO_NOT_USE_THIS
 518         r = p[0]+z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))));
 519         s = one+z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))));
 520 #else
 521         r1 = p[0]+z*p[1]; z2=z*z;
 522         r2 = p[2]+z*p[3]; z4=z2*z2;
 523         r3 = p[4]+z*p[5]; z6=z4*z2;
 524         r= r1 + z2*r2 + z4*r3;
 525         s1 = one+z*q[0];
 526         s2 = q[1]+z*q[2];
 527         s3 = q[3]+z*q[4];
 528         s = s1 + z2*s2 + z4*s3 +z6*q[5];
 529 #endif
 530         return (-.125 + r/s)/x;
 531 }